An algorithm for locating all zeros of a real polynomial

Abstract

We present a globally convergent algorithm for calculating all zeros of a polynomialpn,pn(z) = ∑v = 0navzv, with real coefficients. Splittingpn(exp(it)) into its real and imaginary part we can decide via Euclidean division of Chebyshev expansions and Sturm sequence argumentations whetherpn has some zeros on the unit circle and how many zeros lie on the boundary and in the interior of it. Hence, by a bisection strategy we get the moduli of all zeros to a prescribed accuracy, and additionally we find the arguments as real zeros of a low degree polynomial. In this way we generate starting approximations for all zeros which in a final step are refined by an iterative process of higher order of convergence (e.g. Newton's or Bairstow's method).